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Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth
Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media
1. | Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, France |
2. | Mathematical Institute, University of Oxford, OX2 6GG, UK |
We consider the conductivity equation in a bounded domain in $ \mathbb{R}^{d} $ with $ d\geq3 $. In this study, the medium corresponds to a very contrasted two phase homogeneous and isotropic material, consisting of a unit matrix phase, and an inclusion with high conductivity. The geometry of the inclusion phase is so that the resulting Jacobian determinant of the gradients of solutions $ DU $ takes both positive and negatives values. In this work, we construct a class of inclusions $ Q $ and boundary conditions $ \phi $ such that the determinant of the solution of the boundary value problem satisfies this sign-changing constraint. We provide lower bounds for the measure of the sets where the Jacobian determinant is greater than a positive constant (or lower than a negative constant). Different sign changing structures where introduced in [
Erratum: The name of the second author has been corrected from Haun Chen Yang Ong to Shaun Chen Yang Ong. We apologize for any inconvenience this may cause.
References:
[1] |
G. S. Alberti and Y. Capdeboscq, Lectures on Elliptic Methods for Hybrid Inverse Problems, vol. 25 of Cours Spécialisés, Société Mathématique de France, 2018. |
[2] |
G. Alessandrini and R. Magnanini,
Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.
doi: 10.1137/S0036141093249080. |
[3] |
G. Alessandrini and V. Nesi,
Quantitative estimates on jacobians for hybrid inverse problems, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8 (2015), 25-41.
doi: 10.14529/mmp150302. |
[4] |
G. Alessandrini and V. Nesi,
Univalent $\sigma$-harmonic mappings, Arch. Ration. Mech. Anal., 158 (2001), 155-171.
doi: 10.1007/PL00004242. |
[5] |
H. Ammari, E. Bonnetier, F. Triki and M. Vogelius, Elliptic estimates in composite media with smooth inclusions: An integral equation approach, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 453–495.
doi: 10.24033/asens.2249. |
[6] |
H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, vol. 1846 of Lecture Notes in Mathematics, Springer, 2004.
doi: 10.1007/b98245. |
[7] |
E. S. Bao, Y. Y. Li and B. Yin,
Gradient estimates for the perfect conductivity problem, Archive for Rational Mechanics and Analysis, 193 (2009), 195-226.
doi: 10.1007/s00205-008-0159-8. |
[8] |
L. Berlyand and H. Owhadi,
Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for Rational Mechanics and Analysis, 198 (2010), 677-721.
doi: 10.1007/s00205-010-0302-1. |
[9] |
M. Briane and G. W. Milton,
Change of sign of the correctors determinant for homogenization in three-dimensional conductivity, Archive for Rational Mechanics and Analysis, 173 (2004), 133-150.
doi: 10.1007/s00205-004-0315-8. |
[10] |
Y. Capdeboscq, On a counter-example to quantitative jacobian bounds, Journal de l'École polytechnique - Mathématiques, 2 (2015), 171–178.
doi: 10.5802/jep.21. |
[11] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 2nd edition, Springer Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-03537-5. |
[12] |
S. Friedland,
Variation of tensor powers and spectra, Linear and Multilinear Algebra, 12 (1982/83), 81-98.
doi: 10.1080/03081088208817475. |
[13] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. |
[14] |
H. Kang, K. Kim, H. Lee, J. Shin and S. Yu, Spectral properties of the Neumann-Poincaré
operator and uniformity of estimates for the conductivity equation with complex coefficients,
J. Lond. Math. Soc. (2), 93 (2016), 519–545.
doi: 10.1112/jlms/jdw003. |
[15] |
R. S. Laugesen,
Injectivity can fail for higher-dimensional harmonic extensions, Complex Variables Theory Appl., 28 (1996), 357-369.
doi: 10.1080/17476939608814865. |
[16] |
Y. Y. Li and M. S. Vogelius,
Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal, 153 (2000), 91-151.
doi: 10.1007/s002050000082. |
[17] |
A. D. Melas,
An example of a harmonic map between Euclidean balls, Proc. Amer. Math. Soc., 117 (1993), 857-859.
doi: 10.1090/S0002-9939-1993-1112497-9. |
[18] |
K.-O. Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand., 21 (1967), 17–37 (1968).
doi: 10.7146/math.scand.a-10841. |
[19] |
J. C. Wood, Lewy's theorem fails in higher dimensions, Math. Scand., 69 (1991), 166 (1992).
doi: 10.7146/math.scand.a-12375. |
show all references
References:
[1] |
G. S. Alberti and Y. Capdeboscq, Lectures on Elliptic Methods for Hybrid Inverse Problems, vol. 25 of Cours Spécialisés, Société Mathématique de France, 2018. |
[2] |
G. Alessandrini and R. Magnanini,
Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.
doi: 10.1137/S0036141093249080. |
[3] |
G. Alessandrini and V. Nesi,
Quantitative estimates on jacobians for hybrid inverse problems, Vestnik YuUrGU. Ser. Mat. Model. Progr., 8 (2015), 25-41.
doi: 10.14529/mmp150302. |
[4] |
G. Alessandrini and V. Nesi,
Univalent $\sigma$-harmonic mappings, Arch. Ration. Mech. Anal., 158 (2001), 155-171.
doi: 10.1007/PL00004242. |
[5] |
H. Ammari, E. Bonnetier, F. Triki and M. Vogelius, Elliptic estimates in composite media with smooth inclusions: An integral equation approach, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 453–495.
doi: 10.24033/asens.2249. |
[6] |
H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, vol. 1846 of Lecture Notes in Mathematics, Springer, 2004.
doi: 10.1007/b98245. |
[7] |
E. S. Bao, Y. Y. Li and B. Yin,
Gradient estimates for the perfect conductivity problem, Archive for Rational Mechanics and Analysis, 193 (2009), 195-226.
doi: 10.1007/s00205-008-0159-8. |
[8] |
L. Berlyand and H. Owhadi,
Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for Rational Mechanics and Analysis, 198 (2010), 677-721.
doi: 10.1007/s00205-010-0302-1. |
[9] |
M. Briane and G. W. Milton,
Change of sign of the correctors determinant for homogenization in three-dimensional conductivity, Archive for Rational Mechanics and Analysis, 173 (2004), 133-150.
doi: 10.1007/s00205-004-0315-8. |
[10] |
Y. Capdeboscq, On a counter-example to quantitative jacobian bounds, Journal de l'École polytechnique - Mathématiques, 2 (2015), 171–178.
doi: 10.5802/jep.21. |
[11] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, 2nd edition, Springer Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-03537-5. |
[12] |
S. Friedland,
Variation of tensor powers and spectra, Linear and Multilinear Algebra, 12 (1982/83), 81-98.
doi: 10.1080/03081088208817475. |
[13] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. |
[14] |
H. Kang, K. Kim, H. Lee, J. Shin and S. Yu, Spectral properties of the Neumann-Poincaré
operator and uniformity of estimates for the conductivity equation with complex coefficients,
J. Lond. Math. Soc. (2), 93 (2016), 519–545.
doi: 10.1112/jlms/jdw003. |
[15] |
R. S. Laugesen,
Injectivity can fail for higher-dimensional harmonic extensions, Complex Variables Theory Appl., 28 (1996), 357-369.
doi: 10.1080/17476939608814865. |
[16] |
Y. Y. Li and M. S. Vogelius,
Gradient estimates for solutions of divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal, 153 (2000), 91-151.
doi: 10.1007/s002050000082. |
[17] |
A. D. Melas,
An example of a harmonic map between Euclidean balls, Proc. Amer. Math. Soc., 117 (1993), 857-859.
doi: 10.1090/S0002-9939-1993-1112497-9. |
[18] |
K.-O. Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand., 21 (1967), 17–37 (1968).
doi: 10.7146/math.scand.a-10841. |
[19] |
J. C. Wood, Lewy's theorem fails in higher dimensions, Math. Scand., 69 (1991), 166 (1992).
doi: 10.7146/math.scand.a-12375. |
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